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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">C</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> holds <br/> ( <font color="Olive" title="b1">f</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  iff  <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Olive" title="b1">f</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">C</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  iff  <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c2">f</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  )</span><br/>



<a NAME="E1:57"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
 <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#D4" title="COMPLEX1:def.4">COMPLEX1:def 4</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:57"/>
( <font color="Maroon" title="c2">f</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  implies  <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c2">f</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c2">f</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  implies <font color="Maroon" title="c2">f</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:57_1"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
 <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#D4" title="COMPLEX1:def.4">COMPLEX1:def 4</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:57_1"/>
<font color="Maroon" title="c2">f</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c2">f</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> </span><br/>

<span class="kw">hence </span><a NAME="E3:57_1"/>
 <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c2">f</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="cfunct_1.html#T59" target="_self" title="CFUNCT_1:th.59">Th71</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:57"/>
 <a href="valued_1.html#K31" title="VALUED_1:func.31">-</a> <font color="Maroon" title="c2">f</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> </span><br/>

<span class="kw">hence </span><a NAME="E4:57"/>
<font color="Maroon" title="c2">f</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="cfunct_1.html#T59" target="_self" title="CFUNCT_1:th.59">Th71</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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